Optimal control with reset-renewable resources

TL;DR

This paper develops methods for optimal control problems with resource constraints that can be reset upon entering certain states, providing numerical solutions and algorithms for both discrete and continuous cases, with applications in path planning.

Contribution

It introduces a novel framework for control problems with resettable resources, deriving augmented PDEs and efficient algorithms for their numerical solution.

Findings

Demonstrates the effectiveness of the methods on path planning examples.

Provides algorithms for both extended and original state spaces.

Shows improved computational efficiency in solving reset-resource control problems.

Abstract

We consider both discrete and continuous control problems constrained by a fixed budget of some resource, which may be renewed upon entering a preferred subset of the state space. In the discrete case, we consider both deterministic and stochastic shortest path problems with full budget resets in all preferred nodes. In the continuous case, we derive augmented PDEs of optimal control, which are then solved numerically on the extended state space with a full/instantaneous budget reset on the preferred subset. We introduce an iterative algorithm for solving these problems efficiently. The method's performance is demonstrated on a range of computational examples, including the optimal path planning with constraints on prolonged visibility by a static enemy observer. In addition, we also develop an algorithm that works on the original state space to solve a related but simpler problem: finding the subsets of the domain "reachable-within-the-budget". This manuscript is an extended version of the paper accepted for publication by SIAM J. on Control and Optimization. In the journal version, Section 3 and the Appendix were omitted due to space limitations.